In this text we will focus solely on the B-side of mirror symmetry and consider another enumerative result: the computation of the numbers of disks with boundary on the real quintic Lagrangian inside the quintic threefold. These numbers were first predicted in [1]} and then fully computed in [2]}. This computation relies on the open Gromov-Witten invariants which are also computed in these articles. Instead of just considering the mirror family, we need to fix, in each one of the elements of the family, a pair of homologous rational curves \(C_{\pm }\)  given by specific equations given in (REF ). See Section REF  for details. We refer to [3]} for the reasons to consider these two curves. In this context, we have a new natural period to compute, which is the integral of a holomorphic three form over the homology connecting the two curves above. This period satisfy a non homogenous version of the Picard-Fuchs equation, which can be found on Section REF . In order to execute the ideas from the GMCD program, we consider a relative version of the algebraic de Rham cohomology \(H_{{\rm dR}}^3(W,C_+\cup C_-)\)  and a relative version of the Gauss-Manin connection. We define a moduli space \(\sf T_{op}\)  of triples \((W,C_{\pm }, [\alpha _0,\dots ,\alpha _4])\) , where \(W\)  is an element of the mirror family, \(C_{\pm }\)  is the pair of curves described above on each element of the family and \([\alpha _0,\dots ,\alpha _4]\)  is a basis of the third relative algebraic de Rham cohomology which respect the mixed Hodge structure (for details on mixed Hodge structures, we refer to [4]} and Section REF  of this text). In this case, the Hodge filtration consists of \(F^0\supset F^1\supset \dots \supset F^4 =0\)  with \(\dim F^0 = 5\) , \(\dim F^1=4\) , \(\dim F^2 = 2\) , \(\dim F^3 = 1\)  and the weight filtration consists of \(0 = W_0\subset W_1\subset \dots W_3\) , with \(\dim W_0 = \dim W_1 = 0\) , \(\dim W_2 = 1\)  and \(\dim W_3 = 5\) . These dimensions are computed in Section REF . The idea of considering mixed Hodge structures and relative cohomology in the GMCD framework was examined in the case of elliptic curves with two fixed points in the paper [5]}, in which the authors recovered the theory of Jacobi forms of index zero, and in the paper [6]}, where the authors considered affine Calabi-Yau varieties.
